Is it possible to develop a lazy eye without being born with it?

Yes. Usually this happens if the vision is affected, and a sensory exo/esotropia develops. The eye can also turn in or out in disease states due to muscle restriction, neuro under or over input, muscle hyperactive or weakness.
Yes. Some causes of amblyopia such as strabismus or cataract are not present to any significant degree at birth but progress and become evident as the patient becomes older. An early eye exam can help identify and treat the cause of amblyopia.